1
Charging of Ellipsoidal Monomers
Kevin Bombardier

Abstract—Modeling the charging and aggregation of
monomers in dust aggregates in a plasma helps in understanding
processes such as planetary formation. Many dust charging
models assume the monomers to be spherical. However, there is
evidence to suggest that dust in some astrophysical environments
is not exactly spherical; a better approximation would be to
model ellipsoidal monomers.
In many astrophysical
environments, grains are also charged from collections of plasma
species or radiation. This paper discusses an algorithm used to
calculate the charge on ellipsoidal monomers and compare the
results to spherical monomers. Using a modified version of
orbital motion-limited (OML) theory, the current density to the
surface of a grain can be computed to approximate the charge on
a given monomer. The current to each monomer’s surface is
calculated using a line of sight (LOS) approximation. The LOS
approximation assumes that only the paths from a given point on
a monomer that are not blocked by another monomer contribute
to the charging of that monomer. This current is then used to
calculate the charge and dipole moment. The charge and dipole
moment on an ellipsoidal monomer can be compared to that of a
spherical monomer to better understand the coagulation process.
Index Terms—Dust coagulation,
planetestimal formation, charging.
ellipsoidal
symmetry [2]. By modeling dust grains to be ellipsoidal, the
impact that the charge has on dust coagulation can be better
understood through careful study. Having modeled the
charging process of ellipsoidal monomers, it is possible to
compare the charging of spherical monomers to that of
ellipsoidal monomers. The charge influences the interaction
between dust monomers, and this results in a different overall
structure (or fractal dimension) for dust aggregates [2].
The charge distribution and the dipole moment on a
single monomer are also of interest. For a single spherical
monomer, the charge distribution is symmetric around the
monomer. There is no net dipole for a single spherical
monomer. This differs from an ellipsoidal monomer's charge
distribution and dipole moment. This piece of information is
of particular importance because a dipole moment would
affect the overall evolution of an aggregate.
This paper discusses a modified version of
OML_LOS code for the charging of dust aggregates which
assumes that each monomer to be ellipsoidal rather than
spherical. Results for the charge, charge distribution, and the
dipole moment for an ellipsoidal monomer is also discussed.
monomers,
II. METHOD
I. INTRODUCTION
T
HE coagulation of dust grains is an important area of
research in several fields, especially in the study of
planetesimal formation where dust coagulation is thought
to be the first step in planetary formation [1]. When immersed
in a plasma or radioactive environment, these dust grains
acquire a charge that influences the formation of dust
aggregates. Thus, it is crucial to be able to understand the
charge on these aggregates to better understand planetary
formation.
Numerous studies have modeled the dust grains as
spherical monomers [2],[3]. However, the dust grains are not
always spherical. Previous research has shown differences in
aggregate morphology [4]. However, these models only
considered ballistic collisions between mutual grains. As
noted above, grains in astrophysical environments can be
charged. The case of calculating the charge on these ellipsoids
is a much more difficult task due to the loss of spherical
Manuscript received August 2011. This work was supported in part by the
National Science Foundation under the Research Experience for
Undergraduates program.
Kevin Bombardier did research with Baylor University (e-mail:
[email protected])
Lorin S. Matthews is with the Physics department at Baylor University,
Waco, TX 76798 (email: [email protected]).
The code used to model the charging of dust particles
is based on the OML_LOS code described in [2],[3].
Modifications were made to the code to take into account
ellipsoidal geometry. The algorithm calculates the charge of
the monomers using a modified orbital motion limited (OML)
theory with a line of sight (LOS) approximation.
A. Modified OML Theory
In orbital motion limited theory, the current density to
any given point on monomer α is given by
J=
n  q  fv cos d 3 v
(1)
where n  is the number density of plasma species  far
away from a monomer's potential well, q is the charge of the
plasma species, f is the distribution function,  is the angle
between the velocity vector and the surface normal vector, and
3
this integral is over the velocity space d v that does not
include "blocked" orbits [2]. The distribution function is
assumed to be Maxwellian and is given by
2
because the unit normal vector for a point on a unit sphere is
the vector pointing from the center to the point itself. Since
the vector to a point on a sphere is also its normal vector, the
dot product of each vector direction can be computed to find
all possible cosine factors. However, for an ellipsoid, this is
not as simple. The approach used to generalize this section of
the code instead calculated the gradient of the equation for an
ellipsoidal given by
x2 / a2 + y2 / b2 + z2 / c2 = 1,
Fig. 1. The lines illustrate the lines of sight for a patch
on a monomer. Dashed lines indicated blocked lines of
sight.
3
2
f=
where
 m 
 m 2 q  
v 

 exp  

kT 
 2 kT 
 2kT
(2)
T is the temperature of the plasma species, m is the
mass of the plasma species, k is the Boltzmann's constant, and
 is the dust grain's potential [2].
where a, b, and c are the lengths of the semi-axes of the x, y,
and z coordinates respectively. The gradient, calculated at the
center of each patch, is the normal vector of the ellipsoid, so
once this vector is normalized, the cosine of the angles that the
unit normal vectors and the lines of sight makes with each
other can be calculated.
Another major change to the code was in calculating
the surface area of patches on the ellipsoids. For a sphere, it is
relatively simple to arrange the test points such that the surface
area on each patch of the sphere is constant. However, on an
ellipsoid, this is not so simple (if even possible). Instead, the
surface area of each patch was calculated individually. It is
important to note that the modified version of the code used a
numerical integration to find the surface area of these patches.
The surface area is given by
B. Line of Sight Approximation
By making the substitution
d 3 v  v 2 dvd 2
(4)
SA =
(3)
the integral over speed and direction can be separated where
d2Ω is the differential solid angle of orbits that are not blocked
[2]. Any incoming charged particles are assumed to move in a
straight line. The solid angle of integration is found by
defining m × n test directions, where m is the number of angles
of elevation  and n is the number of azimuthal angles, . Any
blocked lines of sight, as shown in Figure 1, are removed from
the solid angle in the integration.
C. Code modifications
The code was modified to calculate the charge and
dipole moment for ellipsoidal monomers rather than spherical
ones. Test points were used to determine which lines of sight
to remove from the integration. The test points were defined
by points on a unit sphere of a 21 by 20 matrix created by
assigning a range in theta (which denoted the azimuth angle)
and a range in phi (which denoted the elevation angle) to use
as lines of sight and to make ellipsoids out of the unit spheres.
These points were used to determine which lines of sight to
remove from the integration. Also, the points were used to
divide the surface of the ellipsoid into patches.
One difference from the original code was in finding
the cosine of the angles between incoming lines of sight and
the unit normal vectors for a surface patch. In a spherical
monomer, one is able to find the cosine of the angle between
each LOS and the unit normal vector rather easily. This is
max
 max
 sin  
min
2 2
2
a b cos   c
2
b
2
2
2
2

2
cos   a sin  sin  d d
 min
(5)
where θ is the azimuth angle,  is the polar angle, and a,b,c
are the lengths of the x,y,z semi-axes respectively [6]. It
should be noted that the surface area of the patches near the
poles is small and not as accurate as the other patches (where
the elevation is ± π/2). However, since these patches are small
compared to the other patches, the overall charge is hardly
affected.
The last main change to the code was in calculating
the LOS factor. The LOS factor was used to calculate the
charge and is part of the LOS approximation discussed earlier.
To find the LOS factor, the free lines of sights were
determined which indicated which points on the ellipsoid were
blocked and which were open. The LOS factor is
LOS _ factor   cos d 
(5)
where  and d are the same as mentioned earlier [5]. The
calculation to approximate the LOS factor is
LOS _ factor =
3
cosmi  SAmi  free _ LOSmi

m i
(6)
where m and i are indices indicating the azimuth and elevation
of a point.
The code was modified to change its approach on
how to determine if a given point on a monomer was inside
another monomer or not. In the case of spheres, the symmetry
of the sphere makes this task simpler because the radius of a
sphere is constant. The original OML_LOS code calculated
the distance from every point on each monomer to the center
of each monomer. If the distance from a point on a monomer
to the center of another monomer was less than the radius of
the other monomer, then that point was inside another
monomer. The LOS would then be completely blocked. The
modification first calculated the distances from the surface
points on each ellipsoidal monomer to the center of each
monomer. Once this was accomplished, these vectors were
converted into each monomer's coordinate system. These
vectors coordinates were plugged into
x2 / a2 + y2 / b2 + z2 / c2  1,
(7)
where a,b,c were the lengths of the x,y,z semi axes
respectively. If the above condition was satisfied (excluding
the monomer where the point is located on), then that surface
point would be inside (or on) another monomer; thus, its LOS
would be completely blocked.
The code estimated which lines of sight were blocked
by the monomer itself. For spheres, the curvature around any
given point is the same. The curvature of a sphere allows for
any angle that is "behind" the tangent plane for a given point
on a sphere to be considered blocked in the integration. For
ellipsoids, this is not exactly the case. However, computing
the angles to eliminate is a much more difficult task. Instead,
the same approach for spheres was used in the code. This
approximation works well for ellipsoids that do not vary
greatly from spheres. It is important to note that ellipsoids
having a long semi-axis compared to its two other semi-axes
will be less accurately modeled using this approach. A better
approach might be to estimate the curvature by nearby points
on the ellipsoid and to find the angles from there.
The last modification was in calculating blocked lines
of sight due to other monomers blocking incoming charged
particles. The code for ellipsoids took each LOS and
converted them to each monomer's coordinate system (the
origin at the center of the monomer). The lines were then
parameterized in every monomer's coordinate system. The
x,y,z components of the parameterized lines were used in
equation (4). This forms a quadratic equation from the
parametric variable. If the solutions were real, then there
exists some point on the line that intersects the ellipsoid.
Hence, that LOS would be blocked. If the solutions were
imaginary, the LOS would be open (see Fig. 2).
D. Charging Conditions
The charge was calculated for a single monomer
assuming that the electron and ion temperature were T e = Ti =
4637 K, as used in past simulations. The plasma number
density was set at n  = 5 x 108 m-3 [2]. Both spheres and
ellipsoids were charged and the results compared.
III. RESULTS
The code calculated the case of spherical monomers
first as to test whether the code worked properly. A spherical
monomer of radius 3 x 10-6 m attained Q = 3.3540 x 10-16 C.
This agrees with the previous version of OML_LOS for
spheres which calculated Q = 3.3539 x 10-16 C. The charge
distribution for a spherical monomer was plotted as shown in
Fig. 3. For a sphere, the charge on a spherical monomer
should be symmetric on any given monomer. The color of the
surface patches indicates the patch charge divided by the
surface area of the patch. The charge on the surface of a
sphere is constant with some small variations at the poles. The
variations are most likely due to the numerical integration of
the surface area of the patches.
B
t
A
v
Fig 2. Vector t will yield imaginary solutions when its
parametric equation is inserted into the equation for
ellipsoid B and is an open LOS. The parameterization
of vector v will yield real solutions and is a blocked
LOS.
Fig. 3. The charge distribution on each patch on the sphere
was plotted. Blue indicates a negative charge. Red
indicates a positive charge. Overall the charge is
symmetric except for some error at the poles as mentioned
above.
4
Fig. 4. Oblique view of the charge distribution on an
ellipsoidal monomer. The colors represent the charge per
surface area on each patch. More surface points were used to
show the results more clearly.
The charge on an ellipsoid with a = 3 × 10-6 m and a
semi-axis ratio a:b:c = 1:1/3:1/3 was charged and obtained a
charge Q = -1.75 x 10-16 C. This value is reasonable
considering the surface area of this ellipsoid is smaller than the
spherical monomer. The charge distribution on the ellipsoidal
monomer is shown in Figs 4 and 5. The negative charge sits
near the end of the semi-major axis for the ellipsoid. The
positive charge sits near the semi-minor axes.
The dipole moment for this ellipsoid was also
calculated in the code. This ellipsoid had a dipole moment
vector of (1.0 x 10-22) (-0.2300 , 0.0020, 0.0192) which
indicates that the dipole points along the semi-major axis. The
magnitude | p | ≈ 10-22 which is Qd, where d is the length of a
semi-axis.
To further compare the results, a series of ellipsoids
of constant volume and varying length/diameter ratios were
charged and compared. The length is twice the semi-major
axis length and the diameter is twice the semi-minor as used by
Auer (See Fig. 6) [7]. The results were plotted to show the
charge ratio versus the length/diameter ratio. The results were
compared to the shape factor calculated by Auer et al [7] for
varying ellipsoids using the computer program "Coulomb",
Length
Diameter
Fig. 6. The length is the semi-major axis of the
ellipsoid and the diameter is the semi-minor axis of the
ellipsoid.
Fig. 5. Top view of the charge distribution on an ellipsoidal
monomer. The colors represented are the same as in Fig. 3.
Fig. 7. The charge ratio versus the length/diameter ratio at a
constant volume. T he charge increases as the length of the
semi-major axis of the ellipsoid increases.
which solves Maxwell's equations in integral form. Ellipsoids
of varying semi-axis length ratios at a constant volume were
calculated using the modified OML_LOS code. Several
projected data points from Auer et al were used in Fig. 7. In
general, the modified OML_LOS code has a larger charge
ratio than the data in [7]. This may be due to error introduced
when ignoring the curvature of the ellipsoid.
IV. CONCLUSION
This model approximates the charge and dipole
moment on ellipsoidal monomers.
The charge on an
ellipsoidal monomer is larger than that of a spherical monomer
with the same volume. This indicates that the behavior of a
5
charged aggregate with ellipsoidal monomers would differ
from a charged aggregate consisting of spherical monomers.
The charge distribution and the dipole moment on the
ellipsoidal monomer are also of interest. There is a net dipole
on the ellipsoidal monomer whereas the spherical monomer
does not have one due to its symmetry. This may influence
how charged ellipsoidal monomers prefer to collide in contrast
to spherical ones. Ellipsoidal monomers will experience a
torque due to this fact.
It should be noted that because of the approximation
mentioned earlier in calculating which lines of sights are
blocked by the monomer itself, the charges obtained may
differ from their exact value. This approximation becomes
less accurate as the magnitude of the ratio between the semimajor and the semi-minor axes increases.
V. FUTURE WORK
.
By neglecting the curvature of the ellipsoid in
determining what lines of sights are blocked by the monomer
itself, an unwanted error is introduced into the model.
Approximating the blocked lines of sight by the monomer
itself more accurately would be preferred. A possible solution
to this problem would be to estimate the curvature around a
point by using nearby points to determine the blocked angles
(See Fig. 8).
Fig. 8. A possible solution to better estimate the
blocked lines of sight due to the monomer’s own
surface. Nearby points could be used to approximate
the blocked angles.
This model will be applied by building charged
aggregates from ellipsoidal monomers and compare the results
to neutral aggregates as well as charged aggregates built from
spherical monomers to better understand how charging affects
an aggregate's evolution and shape.
ACKNOWLEDGMENT
I would like to thank Lorin S. Matthews and Jonathan
Perry for all their help throughout this project. Thank you to
everyone at CASPER who have been helpful throughout this
experience. This work was funded by NSF grant PHY1002637.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
L.S. Matthews, R.L. Hayes, M.S. Freed, and T.W. Hyde, “Formation of
Cosmic Dust Bunnies,” IEEE Transactions on Plasma Science, vol. 35,
no. 2, pp. 260-264, Apr. 2007.
L.S. Matthews and T.W. Hyde, “Charging and Growth of Fractal Dust
Grains,” IEEE Transactions on Plasma Science, vol. 36, no. 1, pp. 310314, Feb. 2008.
L .S. Matthews and T.W. Hyde, “Effect of dipole-dipole charge
interactions on dust coagulation,” New J. Phys. vol. 11, 2009.
J.D. Perry, E. Gostomski, L.S. Matthews, and T.W. Hyde, “The
Influence of Monomer Shape on Aggregate Morphologies,” Astronomy
and Astrophysics, 2011.
M. Qianyu, L. Matthews, V. Land, and T.W. Hyde, " Charging of
interstellar dust grains near the heliopause," Submitted to Astrophysical
Journal, July, 2011.
Weisstein, Eric W. "Ellipsoid." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/Ellipsoid.html
S. Auer, S. Kempf, and E. Grun, "Computed electric charges of grains
with highly irregular shapes," Workshop on Dust in Planetary Systems.
pp. 177-180, Jan 2007.